Differential Harnack Estimates for Parabolic Equations

Transcription

1 Differenial Harnack Esimaes for Parabolic Equaions Xiaodong Cao and Zhou Zhang Absrac Le M,g be a soluion o he Ricci flow on a closed Riemannian manifold In his paper, we prove differenial Harnack inequaliies for posiive soluions of nonlinear parabolic equaions of he ype f = f f ln f + R f We also commen on an earlier resul of he firs auhor on posiive soluions of he conjugae hea equaion under he Ricci flow 1 Inroducion Le M,g, [0,T, be a soluion o he Ricci flow on a closed manifold M In he firs par of his paper, we deal wih posiive soluions of nonlinear parabolic equaions on M We esablish Li-Yau ype differenial Harnack inequaliies for such posiive soluions More precisely, g evolves under he Ricci flow g = Rc, 1 where Rc denoes he Ricci curvaure of g We firs assume ha he iniial meric g0 has nonnegaive curvaure operaor, which implies ha for all ime [0,T, g has nonnegaive curvaure operaor for example, in he case ha dimension is Xiaodong Cao Deparmen of Mahemaics, Cornell Universiy, Ihaca, NY , USA, cao@mahcornelledu Zhou Zhang Deparmen of Mahemaics, Universiy of Michigan, a Ann Arbor, MI 48109, USA, zhangou@umichedu 1

2 Xiaodong Cao and Zhou Zhang 4, see [7] Consider a posiive funcion f x, defined on M [0,T, which solves he following nonlinear parabolic equaion, f = f f ln f + R f, where he symbol sands for he Laplacian of he evolving meric g and R is he scalar curvaure of g For simpliciy, we omi g in he above noaions All geomery operaors are wih respec o he evolving meric g Differenial Harnack inequaliies were originaed by P Li and S-T Yau in [1] for posiive soluions of he hea equaion herefore also known as Li-Yau ype Harnack esimaes The echnique was hen brough ino he sudy of geomeric evoluion equaion by R Hamilon for example, see [8] and has ever since been playing an imporan role in he sudy of geomeric flows Applicaions include esimaes on he hea kernel; curvaure growh conrol; undersanding he ancien soluions for geomeric flows; proving noncollapsing resul in he Ricci flow [17]; ec See [16] for a recen survey on his subjec by L Ni Using maximum principle, one can see ha he soluion for remains posiive along he flow I exiss as long as he soluion for 1 exiss The sudy of he Ricci flow coupled wih a hea-ype or backward hea-ype equaion sared from R Hamilon s paper [9] Recenly, here has been some ineresing sudy on his opic In [17], G Perelman proved a differenial Harnack inequaliy for he fundamenal soluion of he conjugae hea equaion under he Ricci flow In [], he firs auhor proved a differenial Harnack inequaliy for general posiive soluions of he conjugae hea equaion, which was also proved independenly by S Kuang and Q S Zhang in [11] The sudy has also been pursued in [3, 6, 15, 0] Various esimaes are obained recenly by M Baileseanu, A Pulemoov and he firs auhor in [1], and by S Liu in [13] For nonlinear parabolic equaions under he Ricci flow, local gradien esimaes for posiive soluions of equaion f = f + a f ln f + b f, where a and b are consans, have been sudied by Y Yang in [19] For general evolving merics, similar esimae has been obained by A Chau, L-F Tam and C Yu in [4], by S-Y Hsu in [10], and by J Sun in [18] In [14], L Ma proved a gradien esimae for he ellipic equaion In, if one defines f + a f ln f + b f = 0 ux, = ln f x,, hen he funcion u = ux, saisfies he following evoluion equaion

3 Differenial Harnack Esimaes for Parabolic Equaions 3 u = u u R u 3 The compuaion from o 3 is sandard, which also gives he explici relaion beween hese wo equaions Our moivaion o sudy under he Ricci flow comes from he geomeric inerpreaion of 3, which arises from he sudy of expanding Ricci solions Recall ha given a gradien expanding Ricci solion M,g saisfying R i j + i j w = 1 4 g i j, where w is called solion poenial funcion, we have Rg + g w = n 4 In sigh of his, by aking covarian derivaive for he solion equaion and applying he second Bianchi ideniy, one can see ha Rg + g w g + w = consan Also noice ha he Ricci solion poenial funcion w can be differed by a consan in he above equaions So by choosing his consan properly, we have Rg + g w g = w n 8 One consequence of he above ideniies is he following g w g = g w g w g Rg w 4 Recall ha he Ricci flow soluion for an expanding solion is g = c φ g cf [5], where c = 1 + and he family of diffeomorphism φ saisfies, for any x M, φx = 1 c gwφx Thus he corresponding Ricci solion poenial φ w saisfies φ w x = 1 c gwwφx = φ w x Along he Ricci flow, 4 becomes φ w = φ w φ w R φ w c Hence he evoluion equaion for he Ricci solion poenial is

4 4 Xiaodong Cao and Zhou Zhang φ w = φ w φ w R φ w c 5 The second nonlinear parabolic equaion ha we invesigae in his paper is u = u u R u Noice ha 3 and 6 are closely relaed and only differ by heir las erms Our firs resul deals wih and 3 Theorem 1 Le M,g, [0,T, be a soluion o he Ricci flow on a closed manifold, and suppose ha g0 and so g has weakly posiive curvaure operaor Le f be a posiive soluion o he hea equaion, u = ln f and H = u u 3R n 7 Then for all ime 0,T H n 4 Remark 1 The resul can be generalized o he conex of M being non-compac In order for he same argumen o work, we need o assume ha he Ricci flow soluion g is complee wih he curvaure and all he covarian derivaives being uniformly bounded and he soluion u and is derivaives up o he second order are uniformly bounded in he space direcion Our nex resul deals wih 6, which is also a naural evoluion equaion o consider wih, by he previous moivaion Theorem Le M,g, [0,T, be a soluion o he Ricci flow on a closed manifold, and suppose ha g0 and so g has weakly posiive curvaure operaor Le u be a smooh soluion o 6, and define H = u u 3R n 8 Then for all ime 0,T H 0 Remark If f is a posiive funcion such ha f = e u, hen f saisfies he following evoluion equaion f = f + R f f ln f 1 + In [], he firs auhor sudied he conjugae hea equaion under he Ricci flow In paricular, he following heorem was proved

5 Differenial Harnack Esimaes for Parabolic Equaions 5 Theorem 3 [, Theorem 36] Le M,g, [0,T ], be a soluion o he Ricci flow, and suppose ha g has nonnegaive scalar curvaure Le f be a posiive soluion of he conjugae hea equaion f = f + R f Se v = ln f n ln4πτ, τ = T and Then we have P = v v + R n τ τ P = P P v v i j + R i j 1 τ g i j τ P v R τ τ 9 Moreover, for all ime [0, T, P 0 In he las secion, we apply a similar rick as in he proof of Theorem 1 and obain a slighly differen resul, where we no longer needs o assume ha g has nonnegaive scalar curvaure Proof of Theorem 1 and Applicaion The evoluion equaion of u is very similar o wha is considered in [3] So he compuaion for he very general seing here can be applied Proof Theorem 1 In sigh of he definiion of H from 8 and comparing wih [3, Corollary ], we have u = u u R + R i j u i j u, u = u u u u u R u In fac, one can direcly apply he compuaion resul here wih he only modificaion because of he exra erms coming from ime derivaive u, which are pu a he end of he righ hand side in he above equaliies Then we have H = H H u u i j R i j 1 g i j H u 10 R + R + R u + R i ju i u j u + u,

6 6 Xiaodong Cao and Zhou Zhang where he las wo erms of he righ hand side coming from he exra erm u in 3 Plugging in u + u = H + u 3R n, one arrives a H = H H u u i j R i j 1 g i j + 1 u 3R n + 1 H 11 R + R + R u + R i ju i u j In sigh of he definiion of H 8, for small enough, we have H < 0 Since g i j has weakly posiive curvaure operaor, by he race Harnack inequaliy for he Ricci flow proved by R Hamilon in [8], we have R + R + R u + R i ju i u j 0 Also we have R 0 Noice ha he erm 1 u prevens us from obaining an upper bound for H for > We can deal wih his by he following simple manipulaion To begin wih, one observes ha from he definiion of H, u = u R n H R We also have he following equaliy from definiion, r u i j R i j 1 g i j = u R n Now we can coninue he compuaion for he evoluion of H as follows, H H H u u i j R i j 1 g i j + 1 H u 4R + u R n H n u R n H n + 1 H u H H u n 4R + u R n = H H u + H u 4R n u R n n n n + H H u + H u 4R n + n

7 Differenial Harnack Esimaes for Parabolic Equaions 7 The essenial sep is he second inequaliy where we make use of he elemenary inequaliy u i j R i j 1 g i j 1 u R n n Now we can apply maximum principle The value of H for very small posiive is clearly very negaive So we only need o consider he maximum value poin is a > 0 for he desired esimae For T 0 < T, assume ha he maximum in 0,T 0 ] is aken a 0 > 0 A he maximum value poin, using he nonnegaiviy of u and R, one has H 4n + n 0 = n 1 5 n T + 1 So if T 4, ie, for ime in [0,4, H 0 In general, we have H n 4 Theorem 1 is hus proved As a consequence of Theorem 1, we have Corollary 1 Le M,g, [0,T, be a soluion o he Ricci flow on a closed manifold, and suppose ha g0 and so g has weakly posiive curvaure operaor Le f be a posiive soluion o he hea equaion f = f f ln f + R f Assume ha x 1, 1 and x,, 0 < 1 <, are wo poins in M 0,T Le Γ = inf e γ + R + n + n d, γ 1 4 where γ is any space-ime pah joining x 1, 1 and x, Then we have e 1 ln f x 1, 1 e ln f x, + Γ This inequaliy is in he ype of classical Harnack inequaliies The proof is quie sandard by inegraing he differenial Harnack inequaliy We include i here for compleeness Proof Pick a space-ime curve connecing x 1, 1 and x,, γ = x, for [ 1, ] Recall ha ux, = ln f x, Using he evoluion equaion for u, we have

8 8 Xiaodong Cao and Zhou Zhang d u ux, = + u γ d = u u R u + u γ u u R u + γ Now by Theorem 1, we have u = 1 H + u + 3R + n 1 n 4 + u + 3R + n So we have he following esimaion, d d ux, 1 For any space-ime curve γ, we arrives a d d e u e γ + R + n + n 4 γ + R + n + n 4 u 1 Hence he desired Harnack inequaliy is proved by inegraing from 1 o 3 Proof of Theorem In his secion we sudy u saisfying he evoluion equaion 6 originaed from gradien expanding Ricci solion equaion We invesigae he same quaniy H = u u 3R n as in he las secion The evoluion equaion of u, is sill very similar o wha is considered in [3] We have slighly differen erms coming from ime derivaive u when compuing he evoluion equaion saisfied by H Comparing wih [3, Corollary ], we proceed as follows Proof Theorem Direc compuaion gives he following equaion The modificaion from he compuaion of he reference is minor as illusraed in he proof of Theorem 1 H = H H u u i j R i j 1 g i j H u 13 R + R + R u + R i ju i u j + u + u, +

9 Differenial Harnack Esimaes for Parabolic Equaions 9 where he las wo erms of he righ hand side come from he exra erm u 1+ 6 Plugging in u + u = H + u 3R n, one arrives a H = H H u u i j R i j 1 g i j + + u 4n H 6 R + R + R u + R i ju i u j in + R 14 By he definiion of H, for small enough, we have H < 0 Since g has weakly posiive curvaure operaor, by he race Harnack inequaliy for he Ricci flow [8], we have R + R + R u + R i ju i u j 0 Noice ha now he coefficien for u on he righ hand side is + < 0, and we have R 0 So one can conclude direcly from maximum principle ha H 0 4 A Remark on he Conjugae Hea Equaion In his secion we poin ou a simple observaion for [, Theorem 36] The assumpion on scalar curvaure is no needed below We follow he original se-up in [] Over a closed manifold M n, g for [0,T ] is a soluion o he Ricci flow 1, and f, is a posiive soluion of he conjugae hea equaion f = f + R f, 15 where and R are Laplacian and scalar curvaure wih respec o he evolving meric g Noice ha M f,dµ g is a consan along he flow Se v = log f nlog4πτ, where τ = T and define P := v v + R n τ Now we can prove he following resul which is closely relaed o [, Theorem 36] Theorem 4 Le M,g, [0,T ], be a soluion o he Ricci flow on a closed manifold f is a posiive soluion o he conjugae hea equaion 15, and v is defines as above Then we have max M v v + R increases along he Ricci flow

10 10 Xiaodong Cao and Zhou Zhang Proof The exac compuaion in [, Theorem 36] gives P τ = P P v v + Rc 1 τ g τ P τ v τ R Applying he elemenary inequaliy v + Rc 1 τ g 1 n v + R n τ, and noicing ha P + v + R = v + R n, τ we arrive a P τ P P v 1 n P + v + R τ P + v + R = P P v 1 P + v + R + n + n n τ τ Thus if one defines we have P := P + n τ = v v + R, P τ P P v Hence max M v v + R decreases as τ increases, which means ha i increases as increases This concludes he proof Remark 3 Noice ha we do no need o inroduce τ in Theorem 4, bu we keep he noaion here so i is easy o be compared wih [, Theorem 36] Remark 4 Theorem 4 and [, Theorem 36] esimae quaniies differ by n τ Here we do no need o assume nonnegaive scalar curvaure as in [, Theorem 36] Moreover, one can also prove his resul for complee non-compac manifolds wih proper boundness assumpion Acknowledgemens Xiaodong Cao wans o hank he organizers of he conference Complex and Differenial Geomery for heir inviaion and hospialiy Boh auhors wan o express heir graiude o Eas China Normal Universiy, where hey sared his discussion Cao s research is parially suppored by NSF gran DMS and Zhang s research is parially suppored by NSF gran DMS

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